Mathematical models are used in science to better understand and make predictions of real world phemenon. Often they may help answer questions that cannot be answered by empirical data or experiments. When the phenomenon has elements of uncertainty, which are governed by probability laws, the model is called a probability or a stochastic model. Stochastic models are often characterized by observing a real world situation over time Examples of stochastic models in the biomedical sciences have been developed for therapeutic and early detection clinical trials, epidemics of infectious diseases, cell growth This course is an introduction to stochastic processes as applied to the biomedical sciences. Among the topics which will be discussed are: epidemiology models for incidence, prevalence and mortality, backward and forward recurrence times and their relationship to length biased sampling, Poisson processes, birth and death processes, Markov chains and semi-Markov processes.

Instructor. M. Zelen  (617-432-4914 ,617-632- 3013), Harvard University, USA      

zelen@hsph.harvard.edu

An Introduction to Stochastic Processes in Public Health

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     1. Introduction  (1 lecture)

     2. Elements of  Laplace transforms. (1)

     3. Relations between incidence, prevalence and time with disease.   

    Stable disease model, forward and backward          recurrence times, models for chronological time and age. (2)

     4. Poisson processes

    Exponential distribution normalized spacings, Campbell’s theorem, random sums of exponential random variables, counting processes and the exponential distribution, superposition of counting processes, splitting and component processes, non – homogeneous Poisson processes.(2)

      5. Renewal processes

     Definitions, asymptotics, renewal function, equilibrium renewal processes (1)

       6. Birth and death processes

     Pure birth processes (Yule-Furry process), generalization to birth and death processes, relationship to Markov chains, linear birth and death processes.(2)

       7. Markov chains

    Introduction, Chapman-Kolmogorov equations, branching processes, statistical equilibrium, classification of states (2)

        8. Semi-Markov processes (3)

     Master equations, moments, first passage time problems.               

        9. Recurrent and Transient States

        10.  Semi-Markov Processes